Zero biasing and a discrete central limit theorem


We introduce a new family of distributions to approximate P(W∈A)\mathbb {P}(W\in A) for AβŠ‚{...,βˆ’2,βˆ’1,0,1,2,...}A\subset\{...,-2,-1,0,1,2,...\} and WW a sum of independent integer-valued random variables ΞΎ1\xi_1, ΞΎ2\xi_2, ...,..., ΞΎn\xi_n with finite second moments, where, with large probability, WW is not concentrated on a lattice of span greater than 1. The well-known Berry--Esseen theorem states that, for ZZ a normal random variable with mean E(W)\mathbb {E}(W) and variance Var⁑(W)\operatorname {Var}(W), P(Z∈A)\mathbb {P}(Z\in A) provides a good approximation to P(W∈A)\mathbb {P}(W\in A) for AA of the form (βˆ’βˆž,x](-\infty,x]. However, for more general AA, such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, nonnormal distribution which approximates WW in total variation, and a discrete version of the Berry--Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables (cf. Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237--260]), we introduce a new family of discrete distributions and provide a discrete version of the Berry--Esseen theorem showing how members of the family approximate the distribution of a sum WW of integer-valued variables in total variation.Comment: Published at in the Annals of Probability ( by the Institute of Mathematical Statistics (

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